Algebra Examples

$\frac{\sqrt[5]{x^{6}}}{{(x^{2})}^{\frac{2}{5}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[5]{x^{6}}}{{(x^{2})}^{\frac{2}{5}}}$ .

$x^{a} = \frac{\sqrt[5]{x^{6}}}{{(x^{2})}^{\frac{2}{5}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{6}}$ as $x^{\frac{6}{5}}$ .

$x^{a} = \frac{x^{\frac{6}{5}}}{{(x^{2})}^{\frac{2}{5}}}$

Step 3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{a} = \frac{x^{\frac{6}{5}}}{x^{2 (\frac{2}{5})}}$

Step 4

Move $x^{2 (\frac{2}{5})}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{6}{5}} x^{- 2 (\frac{2}{5})}$

Step 5

Multiply $x^{\frac{6}{5}}$ by $x^{- 2 (\frac{2}{5})}$ by adding the exponents.

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Step 5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{a} = x^{\frac{6}{5} - 2 (\frac{2}{5})}$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $- 2 (\frac{2}{5})$ .

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Step 5.2.1.1

Combine $- 2$ and $\frac{2}{5}$ .

$x^{a} = x^{\frac{6}{5} + \frac{- 2 \cdot 2}{5}}$

Step 5.2.1.2

Multiply $- 2$ by $2$ .

$x^{a} = x^{\frac{6}{5} + \frac{- 4}{5}}$

Step 5.2.2

Move the negative in front of the fraction.

$x^{a} = x^{\frac{6}{5} - \frac{4}{5}}$

Step 5.3

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{6 - 4}{5}}$

Step 5.4

Subtract $4$ from $6$ .

$x^{a} = x^{\frac{2}{5}}$

Step 6

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$a = \frac{2}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{2}{5}$

Decimal Form:

$a = 0.4$